A new geometric approach providing the minimum-energy issue for inverse model controlrelated perfect regulation of linear time-invariant multi-input/single-output plants described in the discretetime state-space framework is proposed in the paper. Recent results have shown that the minimum-norm T-inverse does not guarantee the minimum-energy perfect control design, which has been confirmed by heuristic studies only. The new proposal, postulated throughout the manuscript, certifies the potential of nonunique σ-inverse regarding the minimum-energy behavior of inverse model control-based structures. After application of the proposed geometric approach dedicated to some class of state-space systems, we can precisely calculate the total energy of the multivariable perfect control runs. Thus, the analytical new methodology allows to obtain the minimum-energy inverse model control schemes, what constitutes the main accomplishment of the paper. Additionally, the aim of future analytical exploration covering the entire class of right-invertible state-space systems is clearly focused. INDEX TERMS Geometric solution, perfect control, minimum-energy problem, inverses of nonsquare matrices, discrete-time state-space domain, LTI MIMO.

The application of the switching control framework to the perfect control algorithm is presented in this paper. Employing the nonunique matrix inverses, the different closed-loop properties are obtained and further enhanced with possible switching methodology implementation. Simulation examples performed in the MATLAB/Simulink environment clearly show that the new framework can lead to benefits in terms of the control energy, speed, and robustness of the perfect control law. The possibility of transferring the new obtained results to the symmetrical nonlinear plants seems to be immediate.

In this paper, an attempt at the energy optimization of perfect control systems is performed. The perfect control law is the maximum-speed and maximum-accuracy procedure, which allows us to obtain a reference value on the plant’s output just after a time delay. Based on the continuous-time state-space description, the minimum-error strategy is discussed in the context of possible solutions aiming for the minimization of the control energy. The approach presented within this study is focused on the nonunique matrix inverse-originated so-called degrees of freedom being the core of perfect control scenarios. Thus, in order to obtain the desired energy-saving parameters, a genetic algorithm has been employed during the inverse model control synthesis process. Now, the innovative continuous-time procedure can be applied to a wide range of multivariable plants without any stress caused by technological limitations. Simulation examples made in the MATLAB/Simulink environment have proven the usefulness of the new method shown within the paper. In the extreme case, the energy consumption has been reduced by approximately 80% in comparison with the well-known Moore–Penrose inverse.

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